The Stacks project

Remark 20.34.12. Let $f : (X', \mathcal{O}_{X'}) \to (X, \mathcal{O}_ X)$ be a morphism of ringed spaces. Let $Z \subset X$ be a closed subset and $Z' = f^{-1}(Z)$. Denote $f|_{Z'} : (Z', \mathcal{O}_{X'}|_{Z'}) \to (Z, \mathcal{O}_ X|Z)$ be the induced morphism of ringed spaces. For any $K$ in $D(\mathcal{O}_ X)$ there is a canonical map

\[ L(f|_{Z'})^*R\mathcal{H}_ Z(K) \longrightarrow R\mathcal{H}_{Z'}(Lf^*K) \]

in $D(\mathcal{O}_{X'}|_{Z'})$. Denote $i : Z \to X$ and $i' : Z' \to X'$ the inclusion maps. By Lemma 20.34.2 part (2) applied to $i'$ it is the same thing to give a map

\[ i'_* L(f|_{Z'})^* R\mathcal{H}_ Z(K) \longrightarrow i'_*R\mathcal{H}_{Z'}(Lf^*K) \]

in $D_{Z'}(\mathcal{O}_{X'})$. The map of functors $Lf^* \circ i_* \to i'_* \circ L(f|_{Z'})^*$ of Remark 20.28.3 is an isomorphism in this case (follows by checking what happens on stalks using that $i_*$ and $i'_*$ are exact and that $\mathcal{O}_{Z, z} = \mathcal{O}_{X, z}$ and similarly for $Z'$). Hence it suffices to construct a the top horizonal arrow in the following diagram

\[ \xymatrix{ Lf^* i_* R\mathcal{H}_ Z(K) \ar[rr] \ar[rd] & & i'_* R\mathcal{H}_{Z'}(Lf^*K) \ar[ld] \\ & Lf^*K } \]

The complex $Lf^* i_* R\mathcal{H}_ Z(K)$ is supported on $Z'$. The south-east arrow comes from the adjunction mapping $i_*R\mathcal{H}_ Z(K) \to K$ (Lemma 20.34.1). Since the adjunction mapping $i'_* R\mathcal{H}_{Z'}(Lf^*K) \to Lf^*K$ is universal by Lemma 20.34.2 part (3), we find that the south-east arrow factors uniquely over the south-west arrow and we obtain the desired arrow.

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